Zeros and Multiplicity
A free Algebra II lesson from the “Polynomial Graphs” unit, with a worked example and practice problems including step-by-step solutions.
A zero's multiplicity is the exponent on its factor. Odd multiplicity usually crosses the x-axis; even multiplicity usually touches and turns.
What you'll learn
- Identify zeros from factored form
- Use multiplicity to predict crossing or touching
- Connect roots to graph behavior
Worked example
Problem. For f(x) = (x - 2)^2(x + 5), name the zeros and how the graph behaves.
- x - 2 gives zero 2 with multiplicity 2.
- x + 5 gives zero -5 with multiplicity 1.
- Even multiplicity touches; odd multiplicity crosses.
Answer: 2 touches, -5 crosses
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.
Practice problems
1. For f(x) = (x - 4)(x + 1), the zeros are...
Choices: 4 and -1 · -4 and 1 · 4 and 1 · -4 and -1
Show solution
- Set each factor equal to zero.
Answer: 4 and -1
2. A zero with multiplicity 2 usually...
Choices: Touches and turns · Crosses sharply · Is not a zero · Creates no graph behavior
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- Even multiplicity touches.
Answer: Touches and turns
3. For f(x) = (x + 3)^2(x - 6), which zero touches?
Choices: -3 · 6 · 3 · -6
Show solution
- x + 3 has multiplicity 2.
Answer: -3
4. For f(x) = (x - 1)^3, the graph at x = 1 usually...
Choices: Crosses · Touches only · Has no zero · Ends
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- Odd multiplicity crosses.
Answer: Crosses
5. A cubic with positive leading coefficient has right-end behavior that...
Choices: Rises · Falls · Stays flat · Stops at the x-axis
Show solution
- A cubic has odd degree.
- A positive leading coefficient makes the right end rise.
- So the graph rises to the right.
Answer: Rises
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